ENNs [1]} predict the evidence vector for the predicted Dirichlet distribution instead of softmax probability. Given a sample \(i\)  with the input feature \({\bf x}_i \in \mathbb {R}^L\)  and the ground-truth label \({\bf y}_i\) ,
let \(f({\bf x}_i | \Theta )\)  represents the predicted evidence vector predicted by the classifier with parameters \(\Theta \) .
Then the corresponding Dirichlet distribution has
parameters \({\alpha }_i = f({\bf x}_{i} | \Theta ) + 1\) .
The Dirichlet density \(\text{Dir}({\bf p}_i ; {\alpha })\)  is the prior on the Multinomial distribution \(\text{Multi}({\bf y}_i | {\bf p}_i)\) . Then we optimize the following sum of squared loss for classfication:
\(\mathcal {L}(f({\bf x}_i|\Theta ), {\bf y}_i)=\int \frac{\Vert {\bf y}_i - {\bf p}_i\Vert _2^2}{B({\alpha }_i)} \prod _{j =1}^K p_{ij} ^{(\alpha _{ij}-1)} d{\bf p}_i \nonumber \\=\sum _{j=1}^K (y_{ij}^2 - 2 y_{ij}\mathbb {E}[p_{ij}] +\mathbb {E}[p_{ij}^2]).\) 
